Quantum communication complexity of block-composed functions

A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a {\em total} Boolean function in the two-party interactive model. Razborov's result ({\em Izvestiya: Mathematics}, 67(1):145--159, 2002) implies the conjectured negative answer for functions $F$ of the following form: $F(x, y)=f_n(x_1\cdot y_1, x_2\cdot y_2, ..., x_n\cdot y_n)$, where $f_n$ is a {\em symmetric} Boolean function on $n$ Boolean inputs, and $x_i$, $y_i$ are the $i$'th bit of $x$ and $y$, respectively. His proof critically depends on the symmetry of $f_n$. We develop a lower-bound method that does not require symmetry and prove the conjecture for a broader class of functions. Each of those functions $F$ is the ``block-composition'' of a ``building block'' $g_k : \{0, 1\}^k \times \{0, 1\}^k \rightarrow \{0, 1\}$, and an $f_n : \{0, 1\}^n \rightarrow \{0, 1\}$, such that $F(x, y) = f_n( g_k(x_1, y_1), g_k(x_2, y_2), ..., g_k(x_n, y_n) )$, where $x_i$ and $y_i$ are the $i$'th $k$-bit block of $x, y\in\{0, 1\}^{nk}$, respectively. We show that as long as g_k itself is "hard'' enough, its block-composition with an arbitrary f_n has polynomially related quantum and classical communication complexities. For example, when g_k is the Inner Product function with k=\Omega(\log n), the deterministic communication complexity of its block-composition with any f_n is asymptotically at most the quantum complexity to the power of 7.